Lemma 6.88.
Let \(\phi_*\colon T_1 \to T_2\) and \(\psi_*\colon T_2 \to T_3\) be morphisms of topoi and assume that \(\phi_*\) is essential. Let \(U \in T_1\) be an object which has constant shape relative to the composite \(\psi_* \circ \phi_* \colon T_1 \to T_3\). Then the object \(\phi_\sharp(U) \in T_2\) has constant shape relative to \(\psi_*\).
Proof
The shape of \(\phi_\sharp(U) \in T_2\) relative to \(\psi_*\) is \(\psi_\sharp(\phi_\sharp(U)) \in \Pro(T_3)\). Since \(\phi_*\) is essential, we have \(\phi_\sharp(U) \in T_2\), and the shape of \(U\) relative to \(\psi_* \circ \phi_*\) is
\[(\psi \circ \phi)_\sharp(U) \;\simeq\; \psi_\sharp(\phi_\sharp(U)).\]
By assumption, this is pro-constant, so \(\phi_\sharp(U)\) has constant shape relative to \(\psi_*\).